Post #1655826

2024-07-04 06:06 UTC

It&#39;s finally done! Have a 3 part series on the topological topos First, let&#39;s talk about some fundamentals. How do we think of objects in the topos? How do we compute with them? How do they relate to objects in &quot;the real world&quot;? https://grossack.site/2024/07/03/life-in-johnstones-topological-topos Then, since post 1 is loooong, a little palate cleanser. It&#39;s a short post about how we can use the topological topos to more effectively study algebraic structures equipped with a topology. https://grossack.site/2024/07/03/topological-topos-2-algebras Lastly, another long one. Let&#39;s talk seriously about the internal logic of the topological topos. What&#39;s true and why? And what does this truth mean in the real world? This one isn&#39;t for the faint of heart! https://grossack.site/2024/07/03/topological-topos-3-bonus-axioms --- Thanks for all the encouragement, everyone! This was easily the most effort I&#39;ve ever put into... maybe any math project I&#39;ve ever done, haha. But I think it turned out really well ^_^. If you have comments, I would love to hear what you think!

Replies (2)

@johncarlosbaez@mathstodon.xyz 2024-07-04 08:26

@hallasurvivor - this is great! Not at all where my head is at these days, so I won't be able to offer serious feedback... but here's some less serious feedback: "We think of maps 𝑋→2 being Decidable Propositions. These classify the clopen subsets of 𝑋, and thus are quite rare." I might change that to "and thus may be quite rare". Some people, and famously Johnstone, enjoy Stone spaces, which have lots of clopen sets (at least classically). As you probably know, a Stone space is the same as a profinite topological space, and any Boolean algebra gives a Stone space. So while they're not the kind of space most topologists like, they're great for logicians and even Galois theorists.

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@MartinEscardo@mathstodon.xyz 2024-07-04 17:56

@hallasurvivor Very nice! Thanks for the ack. If I did anything, it was to try to encourage you to complete this project when it was getting hard. 🙂 For the moment I've just had a quick look, because I was preparing my talk the whole day, from 8am to 6:30pm, and now I can't even do modus ponens. But it will be a nice read for my trip to Tallinn on Sunday. I have already left a tab open in my browser in preparation for that. Much better than a movie!

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